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The high school basketball squad from Eagle Grove, population 3,700, had traveled 60 miles up Highway 69 in Iowa to play the team from Forest City, population 4,100. It would be the Eagles against the Indians, a hardwood competition in the center of the country. For some people, this is as American as it gets.

At one point during the online streaming of the game last month, two white announcers for a Forest City radio station, KIOW, began riffing on the Hispanic names of some players from the mildly more diverse community of Eagle Grove. “They’re all foreigners,” said Orin Harris, a longtime announcer; his partner, Holly Jane Kusserow-Smidt, a board operator at the station who was also a third-grade teacher, answered: “Exactly.”

For some people, this is as American as it gets.

Mr. Harris then uttered a term occasionally used these days as a racially charged taunt, or as a braying assertion that the country is being taken back from forces that threaten it. That term is, simply, the surname of the sitting American president.

“As Trump would say, go back where they came from,” Mr. Harris said.

“Well, some would say that, yeah,” Ms. Kusserow-Smidt said. “Some days I feel like that, too.”

Last year’s contentious presidential election gave oxygen to hate. An analysis of F.B.I. crime data by the Center for the Study of Hate & Extremism at California State University, San Bernardino, found a 26 percent increase in bias incidents in the last quarter of 2016 — the heart of the election season — compared with the same period the previous year. The trend has continued into 2017, with the latest partial data for the nation’s five most populous cities showing a 12 percent increase.

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Mr. Trump at a campaign event in Sioux City, Iowa, in November 2016. The contentious presidential election gave oxygen to hateful taunts.CreditDamon Winter/The New York Times

In addition, anti-Muslim episodes have nearly doubled since 2014, according to Brian Levin, the director of the center, which he said has also counted more “mega rallies” by white nationalists in the last two years than in the previous 20. “I haven’t seen anything like this during my three decades in the field,” he said.

Peppered among these incidents is a phenomenon distinct from the routine racism so familiar in this country: the provocative use of “Trump,” after the man whose comments about Mexicans, Muslims and undocumented immigrants — coupled with his muted responses to white nationalist activity — have proved so inflammatory. His words have also become an accelerant on the playing field of sports, in his public criticism of black athletes he deems to be unpatriotic or ungrateful.

Officials at Salem State University in Massachusetts discovered hateful graffiti spray-painted on benches and a fence surrounding the baseball field, including “Trump #1 Whites Only USA.” An undocumented immigrant in Michigan reported to the police that two assailants had stapled a note bearing a slur to his stomach after telling him, “Trump doesn’t like you.” A white Massachusetts businessman at Kennedy International Airport in New York was charged with assaulting and menacing an airline worker in a hijab, saying, among other threats: “Trump is here now. He will get rid of all of you.”

In an email, the White House on Friday denounced the use of the president’s name in cases like these. “The president condemns violence, bigotry and hatred in all its forms, and finds anyone who might invoke his or any other political figure’s name for such aims to be contemptible,” Raj Shah, a White House spokesman, said.

Still, it persists. Across the country, students have used the president’s name to mock or goad minority opponents at sporting events. In March, white fans at suburban Canton High School in Connecticut shouted “Trump! Trump! Trump!” as players from Hartford’s Classical Magnet School, which is predominantly black and Latino, took foul shots during a basketball playoff game. They also chanted “He’s our president!”

The visiting players and their chaperones interpreted the chants not as a sudden burst of presidential fealty, but rather as a slyly racist mantra intended to rattle. As if Donald J. Trump was the president of here, in white suburbia, and not there, in the diverse inner city.

“I’m not sure what politics has to do with basketball,” Azaria Porter, then the Classical team’s 16-year-old manager, told The Hartford Courant. “It was just annoying. It was like, O.K., we get it.”

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President Trump spoke in Harrisburg, Pa., in April. Historians point toward the president’s speeches and tweets to explain how his name became coded as a racially charged jeer.CreditAl Drago/The New York Times

For the record, Classical beat Canton.

According to several scholars of American history, the invocation of a president’s name as a jaw-jutting declaration of exclusion, rather than inclusion, appears to be unprecedented. “If you’re hunting for historical analogies, I think you’re in virgin territory,” said Jon Meacham, the author of several books about presidents, including a Pulitzer Prize-winning biography of Andrew Jackson.

Michael Beschloss, a presidential historian, agrees. “If you’re looking at modern presidents, fill in the blank and see if it can be used in the same way,” he said. “You will see it has not. Hoover? Or Eisenhower? Can you imagine a situation like that?”

The jarring use of Mr. Trump’s name began to surface shortly after he declared his candidacy in June 2015. Within a year, educators were reporting incidents in which, as the Inside Higher Ed website put it, “Trump” had become “a kind of taunt, tossed by largely white students at minority opponents during, say, basketball games.”

But it was not confined to high schools like Dallas Center-Grimes in Iowa, where students mocked a basketball team from the more diverse community of Perry with chants of “Trump, Trump, Trump” in February 2016. Colleges and universities were experiencing similar moments.

Nor was it confined to places of learning. In March 2016, for example, video surveillance at a Kwik Shop in Wichita, Kan., showed a white motorcyclist arguing with two college students — one Hispanic, one Muslim — then assaulting one of them before driving off. The victims later said that the man interspersed his racist epithets with: “Trump, Trump, Trump.” (And yes, the name does tend to come in threes, as if the incantation of his name might summon the man himself.)

Shortly after the election, the Southern Poverty Law Center, which monitors hate crimes, published a report called “The Trump Effect: The Impact of the 2016 Election on Our Nation’s Schools.” Based on a survey of more than 10,000 educators, it detailed an increase in incidents involving swastikas, Nazi salutes and Confederate flags.

“Kids saying, ‘Trump won, you’re going back to Mexico,’” wrote a teacher from Kansas. “A black student was blocked from entering his classroom by two white students chanting, ‘Trump, Trump,’” wrote a teacher from Tennessee. “Seventh-grade white boys yelling, ‘Heil Trump!’” wrote a teacher from Colorado.

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An analysis of F.B.I. crime data by the Center for the Study of Hate & Extremism at California State University, San Bernardino, found a 26 percent increase in bias incidents in the last quarter of 2016 — the heart of the election season.CreditDamon Winter/The New York Times

It is a far cry from wearing a button that says “I Like Ike.”

Mr. Beschloss recalled moments in recent American history when, say, the X in President Richard M. Nixon’s name appeared as a swastika, or a caricature of President Lyndon B. Johnson featured a Hitlerian mustache. But these were generally the acts of opponents to those presidents’ policies during the Vietnam War.

“The message here,” Mr. Beschloss said, “is ‘Trump is going to come and get you — and we support that.’”

There have also been cases in which anti-Trump protesters have harassed and assaulted supporters of the president for, say, wearing a “Make America Great Again” cap. In some instances, the name Trump is invoked in punctuation.

When asked how a president’s very name could become so coded, Mr. Beschloss cited Mr. Trump’s speeches and tweets, including two in particular: the announcement of his candidacy in 2015, during which he referred to Mexican immigrants as criminals, drug dealers and rapists; and his equivocating comments after a white supremacist rally and counterprotest in Charlottesville, Va., in June ended with one person killed and 19 wounded. (“We condemn in the strongest possible terms this egregious display of hatred, bigotry and violence on many sides,” the president had said. “On many sides.”)

“This broadened into a feeling by some people — right or wrong — that Trump is going to be a weapon to reduce the opportunities of those who are different,” Mr. Beschloss said. “This is a signal moment.”

Leah Wright Rigueur, an assistant professor of public policy at the John F. Kennedy School of Government at Harvard, agreed, saying that Mr. Trump’s status as a racial wedge was of his own doing.

“When Trump says, ‘I hear you, I will represent you,’ he is speaking to a particular cross-section of the nation that does not include Muslims, that does not include people of color,” Ms. Wright Rigueur said.

Khalil Gibran Muhammad, a professor of race, history and public policy also at the Kennedy School of Government, said that Mr. Trump had created his own breakaway brand, making him the personification of specific ideals.

“To use the name as a rallying cry for a kind of embodied white supremacy, white nationalism or sense of triumphalism, for taking back the country, as best as I can tell has never been crystallized in the name of a U.S. president,” Mr. Muhammad said.

The sporadic episodes — as chronicled by ProPublica’s “Documenting Hate” project, among others — continue. A “Heil Trump” here, the Trump name scrawled beside a swastika there. In late September, two high school football teams in the Salt Lake City suburbs were squaring off when cheers erupted. Someone was brandishing a cardboard cutout of Mr. Trump, and there began the chanting of three words that have electrified some and unnerved others: “Build the wall! Build the wall!”

Back in Iowa, there have been consequences and remorse in the wake of those two Forest City radio announcers musing on the Hispanic names of some of the players from Eagle Grove.

Ms. Kusserow-Smidt, 63, was fired as a board operator for the radio station; she has since resigned from the Forest City School District. Mr. Harris, 76, who had been with the station for more than 40 years, was also fired. The two have expressed deep regret for their comments, which they said did not reflect who they truly were.

The xenophobic words of the two announcers stung some of the Eagle Grove players, including Nikolas Padilla, whose mother is from Iowa and whose father is from Mexico. Mr. Padilla, a 17-year-old senior, said that he briefly considered quitting because he did not want to be singled out for his Mexican heritage.

One particular comment by the broadcasters — “As Trump would say, go back where they came from” — puzzled Nikolas. His mother, Misty, recalled what her teenage son had said:

“Um, I came from Mason City, Iowa.”

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Audra Burch contributed reporting.

A version of this article appears in print on , on Page A1 of the New York edition with the headline: A Rallying Cry or a Racial Taunt Invoking the President: ‘Trump!’. Order Reprints | Today’s Paper | Subscribe